\documentstyle[twoside]{article}
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\markboth{\hfil Least-energy Solutions \hfil EJDE--1993/05}%
{EJDE--1993/05\hfil Xiaofeng Ren \hfil}
\begin{document}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations}\newline
Vol. 1993(1993), No. 05, pp. 1-21. Published October 15, 1993.\newline
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
Least-energy Solutions to a Non-autonomous Semilinear Problem
with Small Diffusion Coefficient
\thanks{\copyright 1993 Southwest Texas State
University and University of North Texas\newline\indent
Submitted: August 19, 1993.\newline\indent
{\em 1991 Mathematics Subject Classifications:} 35B25, 35B40.\newline\indent
{\em Key words and phrases:} Least-energy solution, Spiky pattern.} }
\date{}
\author{Xiaofeng Ren}
\maketitle
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\newcommand{\pf}{{\it Proof. }}
\newcommand{\ep}{$\Box$}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}[lemma]{Proposition}
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{corollary}[lemma]{Corollary}
\newtheorem{remark}[lemma]{Remark}
\begin{abstract}
Least-energy solutions of a non-autonomous semilinear problem with a
small diffusion coefficient are studied in this paper. We prove that
the solutions will develop single peaks as the diffusion coefficient
approaches 0. The location of the peaks is also considered in this
paper. It turns out that the location of the peaks is determined by
the non-autonomous term of the equation and the type of the boundary
condition. Our results are based on fine estimates of the energies
of the solutions and some non-existence results for semilinear
equations on half spaces with Dirichlet boundary condition and some
decay conditions at infinity.
\end{abstract}
\section{Introduction}
\setcounter{equation}{0}
This work is devoted to the
least-energy solutions of a non-autonomous semilinear problem with
a small diffusion coefficient. Considering
\begin{equation}
\left \{ \begin{array}{l}
\epsilon^{2}\Delta u-u+K(x)u^{p}=0 \ \mbox{in} \ \Omega \\
B(u)=0
\end{array} \right. \end{equation}
where
\begin{itemize}
\item $\Omega$ is a smooth bounded domain in $R^{N}$,
\item $K(x)>0$ in $\overline{\Omega}$ is a $C^{\alpha}$ function,
\item $1

1$ if $N=2$,
\item $B(u)$ is the
boundary operator which is either Dirichlet, i.e. $B(u)=u|_{\partial
\Omega}$, or Neumann, i.e. $B(u)=\frac{\partial u}{\partial
\nu}|_{\partial \Omega}$, and
\item $\epsilon$ is a small parameter,
\end{itemize}
we would like to understand the behavior
of positive least-energy solutions to (1.1) as $\epsilon$ tends to
0.
The problem is motivated by some pattern formation problems in
biology. Keller and Siegel in \cite{ks} proposed a model to describe
the chemotactic aggregation stage of cellular slime molds. Let
$u_{1}(x,t)$ be the population of amoebae at place $x$ and time $t$
and let $u_{2}(x,t)$ be the concentration of the chemical. Then a
simplified Keller-Siegel system may be written as
\begin{equation} \left \{ \begin{array}{l}
\displaystyle \frac{\partial u_{1}}{\partial t}=D_{1}\Delta u_{1}-\chi \nabla
(u_{1}\nabla \phi (u_{2})) \ \mbox{in} \ \Omega \times [0,T) \\ \ \\
\displaystyle \frac{\partial u_{2}}{\partial t}=D_{2}\Delta u_{2}+k(x, u_{1}, u_{2})
\ \mbox{in} \ \Omega \times [0,T) \\ \ \\
\displaystyle \frac{\partial u_{1}}{\partial \nu}=\frac{\partial
u_{2}}{\partial
\nu}=0 \ \mbox{on} \ \partial \Omega \times [0, T) \\ \ \\
u_{1}(x,0)=u^{(0)}_{1}(x)>0, \ u_{2}(x,0)=u^{(0)}_{2}(x)>0 \ \mbox{on}
\ \overline{\Omega}.
\end{array} \right. \end{equation}
We refer to \cite{ks} for more information about this model. If we take
$\phi(u_{2})=\log u_{2}$ and $k(x, u_{1}, u_{2})=-au_{2}+b(x)u_{1}$
with $b(x)>0$ on $\overline{\Omega}$, and consider steady states of
the above system, then the system for steady states is reduced to (1.1)
with the Neumann boundary condition. We refer to C. Lin, W.-M. Ni and I.
Takagi \cite{lnt} for the details of the derivation. Numerical
experiments indicate that (1.2) possesses stable steady states with
spiky patterns when $D_{2}$ is small. On the other hand, A. Gierer and
H. Meinhardt discussed a activator-inhibitor problem in
\cite{gm}, yielding the so-called Gierer-Meinhardt system.
Numerical experiments there also indicate
point-condensation phenomena. Under certain conditions the
Gierer-Meinhardt system can be reduced to (1.1) too. We again refer
to \cite{lnt} for the derivation. In this paper we shall show that
the least-energy solutions of (1.1) with either the Dirichlet
boundary condition or the Neumann boundary condition develop
spiky pattern of single peak as $\epsilon$ approaches $0$.
The autonomous case of (1.1), i.e. $K(x)=1$, with the Neumann boundary
condition has been studied by C.
Lin, W.-M. Ni and I. Takagi in a series of papers \cite{lnt},
\cite{nt} and \cite{nt2}. They proved that the least-energy solutions
possesses a single peak on the boundary of $\Omega$ as $\epsilon$ tends
to 0. The problem of locating these peaks is finally settled by W.-M.
Ni and I. Takagi in \cite{nt2} where they proved that the peaks will
approach a point on the boundary of $\Omega$ which assumes the maximum
of the mean curvature of $\partial \Omega$. W.-M. Ni, X. Pan and I.
Takagi also considered the critical case, i.e. $p=\frac{N+2}{N-2}$, in
\cite{npt} and \cite{p} where they showed that the least-energy
solutions will blow up at the boundary as $\epsilon$ tends to 0.
Here we shall focus on the non-autonomous term $K(x)$ and see how it
will affect the shape of the least-energy solutions and the location
of the peaks. We shall see that when $K(x)$ is nontrivial the location
of peaks is indeed dominated by $K(x)$. Therefore in most situations
the effect of $K(x)$ overrides the effect of the geometry of $\Omega$.
Let $H$ be the Hilbert space $W_{0}^{1,2}(\Omega)$ if $B(u)=u|_{\partial
\Omega}$ takes the Dirichlet boundary condition, or $W^{1,2}(\Omega)$ if
$B(u)=\frac{\partial u}{\partial \nu}|_{\partial \Omega}$ takes the
Neumann boundary condition. Define energy
\[ J_{\epsilon}: \ H \rightarrow R \]
by
\begin{equation}
J_{\epsilon}(v)=\frac{1}{2}\int_{\Omega}[\epsilon^{2}|\nabla v|^{2}+v^{2}]dx
-\frac{1}{p+1}\int_{\Omega} K(x)v_{+}^{p+1}dx
\end{equation} where $v_{+}(x)=\max (v(x),0)$. Choose $e\in H$, $e\geq
0$, $e\not =0$, such that $J_{\epsilon}(e)=0$. Then the well known
mountain-pass lemma asserts that there is a nontrivial positive
critical point, i.e. a solution of (1.1), $u_{\epsilon}$ of
$J_{\epsilon}$ with positive critical value $c_{\epsilon}$ such that
\begin{equation}
c_{\epsilon}:=\inf _{h\in \Gamma}\max_{t \in [0,1]}
J_{\epsilon}(h(t))=J_{\epsilon}(u_{\epsilon})
\end{equation} where $\Gamma$ is the class of all paths $h(t)$ in $H$
connecting $0$ and $e$, i.e. $h(0)=0$ and $h(1)=e$. We refer to
\cite{ar} for the mountain-pass lemma. It turns out in
Proposition 2.1 that $u_{\epsilon}$ does not depend on the choice of
$e$ and $u_{\epsilon}$ achieves the least energy among all nontrivial
critical points of $J_{\epsilon}$, so we call these
mountain-pass solutions least-energy solutions.
To investigate the asymptotic behavior of $u_{\epsilon}$, we start
with the energy of $u_{\epsilon}$. Once we obtain the asymptotic
behavior of the energy of $u_{\epsilon}$ in Proposition 3.1 and 3.2
with the aid a so-called ground state profile (see (2.3)),
we shall prove for the Dirichlet problem
\begin{theorem}
Letting $u_{\epsilon}$ be a least-energy solution of (1.1) with the
Dirichlet boundary condition, we have
\begin{enumerate}
\item There exist positive constants $C_{1}$ and $C_{2}$ independent of
$\epsilon$ such that
\[ C_{1} \leq \|u_{\epsilon}\|_{L^{\infty}(\Omega)} \leq C_{2}. \]
\item For $\epsilon$ small enough $u_{\epsilon}$ has only one
local maximum point $P_{\epsilon}$ with
\[ \lim_{\epsilon \rightarrow 0}\epsilon^{-1}\mbox{{\em dist}}
(P_{\epsilon}, \partial \Omega)=\infty. \]
\item If $P$ is a limit point of $\{ P_{\epsilon} \}$ as $\epsilon
\rightarrow 0$, then
\[ K(P)=\max_{x\in \overline{\Omega}}K(x). \]
\end{enumerate}
\end{theorem}
The story for the Neumann problem seems more complicated. Depending on
\[ \max_{x\in \overline{\Omega} }K(x) > 2^{\frac{p-1}{2}}\max_{x\in
\partial \Omega}K(x) \] or \[ \max_{x\in \overline{\Omega}}K(x)
< 2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x), \] we have
\begin{theorem}
Let $u_{\epsilon}$ be a least-energy solution of (1.1) with the
Neumann boundary condition. Assuming
\[ \max_{x\in \overline{\Omega}
}K(x) > 2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x), \]
we have the following:
\begin{enumerate}
\item There exist positive constants $C_{1}$ and $C_{2}$ independent
of $\epsilon$ such that
\[ C_{1} \leq \|u_{\epsilon}\|_{L^{\infty}(\Omega)} \leq C_{2}. \]
\item For $\epsilon$ small enough $u_{\epsilon}$ possesses only one
local maximum point $P_{\epsilon}$ which stays away from the boundary
of $\Omega$ as
$\epsilon \rightarrow 0$.
\item Every limit point of $\{ P_{\epsilon} \}$ as
$\epsilon \rightarrow 0$ must be a maximum point of $K(x)$ in the
interior of $\Omega$.
\end{enumerate}
\end{theorem}
\begin{theorem}
Let $u_{\epsilon}$ be a least-energy solution of (1.1) with the
Neumann boundary condition. Assuming
\[ \max_{x\in \overline{ \Omega}
}K(x) < 2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x), \]
we have the following:
\begin{enumerate}
\item There exist positive constants $C_{1}$ and $C_{2}$ independent
of $\epsilon$ such that
\[ C_{1} \leq \|u_{\epsilon}\|_{L^{\infty}(\Omega)} \leq C_{2}. \]
\item For $\epsilon$ small enough $u_{\epsilon}$ possesses only one
local maximum point on the boundary of $\Omega$,
\item Every limit point of $\{ P_{\epsilon} \}$ as
$\epsilon \rightarrow 0$ must be a maximum point of $K(x)$ restricted
on $\partial \Omega$.
\end{enumerate}
\end{theorem}
We shall also see the shape of $u_{\epsilon}$ in the proofs of these
theorems. I would like to mention that in the works \cite{rw}
\cite{rw2} of J. Wei and myself spiky patterns have also
appeared in some different problems with the parameters in the
exponents.
This article is organized as follows: After some preliminaries in section
2, we shall describe the energy of $u_{\epsilon}$ in
section 3. Then we shall prove Theorem 1.1 in section 4 and Theorem
1.2 and 1.3 in section 5.
\section{Preliminaries}
\setcounter{equation}{0}
We first give an alternative description of the least energy solutions
to (1.1) which implies that the mountain-pass solutions have the
least energy among all nontrivial critical points of
$J_{\epsilon}$. The proof of the equivalence is identical with the proof of
Lemma 3.1 in \cite{nt}. We include it here for completeness.
\begin{lemma}
Letting $u_{\epsilon}$ be a mountain-pass solution of (1.1) with either
the Dirichlet boundary condition or the Neumann boundary condition,
then we have that
$J_{\epsilon}(u_{\epsilon})$
does not depend on the choice of $e$ where $e\geq 0$ with $e\not \equiv 0$ and
$J_{\epsilon}(e)=0$ is the base point in the mountain-pass lemma.
Furthermore, $J_{\epsilon}(u_{\epsilon})$ is the least
positive critical value of $J_{\epsilon}$ and characterized by
\begin{equation}
J_{\epsilon}(u_{\epsilon})=\inf\{M[v]: v\in
H,v\not\equiv
0 \ \mbox{{\em and}} \ v\geq 0\ \mbox{{\em in}}\ \Omega\}
\end{equation}
where
\[ M[v]=\sup_{t \geq 0} J_{\epsilon}(tv)=\max_{t \geq 0} J_{\epsilon}(tv).
\] \end{lemma}
\pf
Let $v\in H$ be non-negative with $|\{x: \ v(x)>0 \}|>0$. Putting
\[ h_{\epsilon}(t):=J_{\epsilon}(tv) \]
for $t>0$. We note that $h_{\epsilon}(t)$ has a unique positive
critical point.
Now fix a non-negative function $e \not \equiv 0$ in $H$ with
$J_{\epsilon}(e)=0$ and apply the mountain-pass lemma to obtain the
critical value $c_{\epsilon}$. Let $u_{\epsilon}$ be a critical point
of $J_{\epsilon}$ with $J_{\epsilon}(u_{\epsilon})=c_{\epsilon}$.
Since $u_{\epsilon}>0$ and $J'_{\epsilon}(u_{\epsilon})=0$, it is clear
that $M[u_{\epsilon}]=c_{\epsilon}$ and hence
\begin{equation} c_{\epsilon} \geq \inf\{M[v]: v\in
H, v\not\equiv
0 \ \mbox{{\em and}} \ v\geq 0\ \mbox{{\em in}}\ \Omega\}.
\end{equation}
Suppose that the strict inequality holds in (2.2). Then there is a
non-negative function $v_{\ast}\not \equiv 0$in $H$ such that
\[ M[v_{\ast}] < c_{\epsilon}. \]
>From the above observation we know that there is $T_{\ast}>0$ such
that $e_{\ast}:=T_{\ast}v_{\ast}$ satisfies
$J_{\epsilon}(e_{\ast})=0$. Consider the set $V^{+}:=\{\lambda e+\mu
e_{\ast}: \ \lambda ,\ \mu \geq 0 \}$ and the two dimensional subspace
$V$ of $H$ spanned by $e$ and $e_{\ast}$. Let $S$ be a circle with
radius $R$ so large that for $R>\max (\|e\|, \|e_{\ast}\|)$ and
$J_{\epsilon}\leq 0$ on $S\cap V^{+}$. Let $\gamma$ be the path
consisting of the line segment with endpoints $0$ and
$Re_{\ast}/\|e_{\ast}\|$, the circular arc $S\cap V^{+}$ and the line
segment with endpoints $Re/\|e\|$ and $e$. Clearly $\gamma$ belongs to
$\Gamma$ defined in (1.4). It is also easy to see that, along $\gamma$,
$J_{\epsilon}$ is positive only on the line segment joining $0$ and
$e_{\ast}$. Hence by the definition of $M[v_{\ast}]$ one finds that
\[ \max_{v\in \gamma}J_{\epsilon}(v)=M[v_{\ast}]0, \ \lim_{|x|\rightarrow \infty}u(x)=0 \ \mbox{and} \ \nabla
u(0)=0
\end{array} \right. \end{equation}
serves as an asymptotic profile for $u_{\epsilon}$. We collect some
well-known facts about $w$. The proofs of these facts can be found in
M. Kwong \cite{k}.
\begin{proposition}
\begin{enumerate}
\item (2.3) has a unique solution in $W^{1,2}(R^{N})$.
\item $w$ is spherical symmetric: $w(z)=w(r)$ with $r=|z|$ and
$\frac{dw}{dr}<0$ for $r>0$.
\item $w$ and its derivatives decay exponentially at infinity,
i.e. there exit positive constants $C$ and $\mu$ such that
$|D^{\alpha}w|\leq Ce^{-\mu |z|}$ for all $z\in R^{N}$
with $|\alpha |\leq 1$.
\end{enumerate}
\end{proposition}
We also adapt a non-existence result of M.
Esteban, and P. Lions \cite{el} for some unbounded domains with a
sketch of their proof for completeness.
\begin{proposition}
Let $\Omega$ be a unbounded smooth domain in $R^{N}$ so
that there exits $X\in R^{N}, |X|=1$, such that $(\nu (x),
X)\geq 0$
and $(\nu (x), X)\not\equiv 0$ where $\nu (x)$ is the outward unit
normal vector at $x \in \partial \Omega$. (Notice: Half space $
(R^{N})^{+}$
satisfies the condition about $\Omega$). Suppose $u$ satisfies
\[ \Delta u+f(u)=0 \]
in $\Omega$, $u \in C^{2}(\overline{\Omega})$, $u=0$ on
$\partial\Omega$ where $f$ is a locally Lipschitz function on $ R^{N}$
with $f(0)=0$. If, in addition, $\nabla u\in L^{2}(\Omega), F(u)\in
L^{1}(\Omega)$ where $F(t)=\int_{0}^{t}f(s)ds$, then we have
necessarily $u\equiv 0$ in $\Omega$.
\end{proposition}
\pf Multiplying the equation by $\frac{\partial u}{\partial
x_{i}}$ and integrating it by parts over $\Omega\cap B_{R}$ where
$B_{R}$ is the ball centered at origin with radius $R$, we obtain
\[ 0=\int_{\Omega\cap B_{R}}[\Delta u+f(u)]\frac{\partial u}{\partial
x_{i}} \]
\[ =\int_{\Omega\cap B_{R}}[\frac{\partial F(u)}{\partial
x_{i}}-(\nabla u, \nabla \frac{\partial u}{\partial
x_{i}})]+\int_{\partial (\Omega\cap B_{R})}\frac{\partial u}{\partial
x_{i}}\frac{\partial u}{\partial \nu} \]
where $\nu$ denotes the outer normal of the domain.
Let $\nu (x)=(\nu_{1}(x),
\nu_{2}(x),...,\nu_{N}(x))$. Integrating by parts and $\nabla
u=\frac{\partial u}{\partial \nu}$ on $\partial \Omega$ yield
\[ \int_{\Omega\cap B_{R}}\frac{\partial F(u)}{\partial
x_{i}}=\int_{\partial (\Omega\cap
B_{R})}F(u)\nu_{i}(x)=\int_{\Omega\cap\partial
B_{R}}F(u)\frac{x_{i}}{|x|}; \]
\[ \int_{\Omega\cap B_{R}}(\nabla u, \nabla \frac{\partial u}{\partial
x_{i}})=\frac{1}{2}\int_{\partial (\Omega\cap B_{R})}|\nabla
u|^{2}\nu_{i}(x);
\]
\[ \int_{\partial(\Omega\cap B_{R})}\frac{\partial u}{\partial
x_{i}}\frac{\partial u}{\partial \nu}=\int_{\partial B_{R}\cap
\Omega}\frac{\partial u}{\partial x_{i}}\frac{\partial u}{\partial
\nu}+\int_{\partial \Omega\cap B_{R}}|\nabla u|^{2}\nu_{i}(x). \]
Hence
\[ -\frac{1}{2}\int_{\partial \Omega \cap
B_{R}}|\nabla u|^{2}\nu_{i}(x)=\int_{\Omega\cap \partial
B_{R}}[F(u)\frac{x_{i}}{x}-\frac{1}{2}|\nabla
u|^{2}\frac{x_{i}}{x}+\frac{\partial u}{\partial x_{i}}\frac{\partial
u}{\partial \nu}]. \]
We then obtain
\[ |\int_{\partial \Omega\cap B_{R}}|\nabla u|^{2}\nu_{i}(x)|\leq
\int_{\partial B_{R}\cap \Omega}[|F(u)|+\frac{3}{2}|\nabla u|^{2}].
\]
Since $\nabla u\in L^{2}(\Omega), F(u)\in L^{1}(\Omega)$,
\[ \int_{0}^{\infty}\int_{\partial \Omega\cap B_{R}}|\nabla
u|^{2}\nu_{i}(x)dsdR\leq \int_{0}^{\infty}\int_{\partial B_{R}\cap
\Omega}[F(u)+\frac{3}{2}|\nabla u|^{2}] \]
\[ =\int_{\Omega}[F(u)+\frac{3}{2}|\nabla u|^{2}]0$ such that
$B_{1}=B_{r_{1}}(O) \subset \Omega \subset
B_{2}=B_{r_{2}}(O)$. Define
\[ \underline{J_{\epsilon}}: \ W^{1,2}_{0}(B_{1})\rightarrow R \]
by
\[
\underline{J_{\epsilon}}(v)=\frac{1}{2}\int_{B_{1}}[\epsilon^{2}|\nabla
v|^{2}+v^{2}]dx - \frac{1}{p+1}\int_{B_{1}}[\min_{x\in
B_{r_{1}}}K(x)]v^{p+1}_{+}dx. \]
Also define
\[ \overline{J_{\epsilon}}: \ W^{1,2}_{0}(B_{2})\rightarrow R \]
by
\[
\overline{J_{\epsilon}}(v)=\frac{1}{2}\int_{B_{2}}[\epsilon^{2}|\nabla
v|^{2}+v^{2}]dx - \frac{1}{p+1}\int_{B_{2}}[\max_{x\in
\overline{\Omega}}K(x)]v^{p+1}_{+}dx. \]
Through trivial extension, we may write
\[ W^{1,2}_{0}(B_{1}) \subset W^{1,2}_{0}(\Omega) \subset
W^{1,2}_{0}(B_{2}). \]
Therefore with the aid of Lemma 2.1 we conclude
\begin{equation}
\overline{J_{\epsilon}}(\overline{u}_{\epsilon}) \leq
J_{\epsilon}(u_{\epsilon}) \leq
\underline{J_{\epsilon}}(\underline{u}_{\epsilon})
\end{equation}
where $\underline{u}_{\epsilon}$ is a least-energy critical point of
$\underline{J_{\epsilon}}$ and $\overline{u}_{\epsilon}$ is a
least-energy critical point of $\overline{J_{\epsilon}}$. Therefore
$\underline{u}_{\epsilon}$ solves
\[ \left \{ \begin{array}{l}
\epsilon^{2}\Delta u-u+[\min_{x\in B_{r_{1}}}K(x)]u^{p}=0 \\
u|_{\partial B_{1}}=0
\end{array} \right. \]
and $\overline{u}_{\epsilon}$ solves
\[ \left \{ \begin{array}{l}
\epsilon^{2}\Delta u-u+[\max_{x\in \overline{\Omega}}K(x)]u^{p}=0 \\
u|_{\partial B_{2}}=0 .
\end{array} \right. \]
Now we can focus on
$\underline{J_{\epsilon}}(\underline{u}_{\epsilon})$ and
$\overline{J_{\epsilon}}(\overline{u}_{\epsilon})$. Let
$\underline{w}_{\epsilon}=\underline{u}_{\epsilon}(\epsilon x+O)$,
$\overline{w}_{\epsilon}=\overline{u}_{\epsilon}(\epsilon x+O)$.
Observe that
$\underline{w}_{\epsilon}$ solves
\begin{equation} \left \{ \begin{array}{l}
\Delta u-u+[\min_{x\in B_{r_{1}}}K(x)]u^{p}=0 \ \mbox{in} \
B_{\epsilon^{-1}r_{1}} \\
u|_{\partial B_{\epsilon^{-1}r_{1}}}=0
\end{array} \right. \end{equation}
and $\overline{w}_{\epsilon}$ solves
\begin{equation} \left \{ \begin{array}{l}
\Delta u-u+[\max_{x\in \overline{\Omega}}K(x)]u^{p}=0 \ \mbox{in} \
B_{\epsilon^{-1}r_{2}} \\
u|_{\partial B_{\epsilon^{-1}r_{2}}}=0.
\end{array} \right. \end{equation}
It is well known that any positive solution to each of the above two equations
is radially symmetric, (see \cite{gnn}).
\ %%%%%%%%%%%%%%%%
\noindent Claim:
\[\underline{w}_{\epsilon} \rightarrow [\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{1}{p-1}}w; \]
\[ \overline{w}_{\epsilon} \rightarrow [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{1}{p-1}}w \]
in $W^{1,2}(R^{N})$ and $C^{2,\alpha}_{loc}(R^{N})$.
\ %%%%%%%%%%%%%%
We shall only prove the convergence for $\underline{w}_{\epsilon}$;
the other part follows in the same
way. From the characterization in Lemma 2.1 we conclude that
\[
I_{\epsilon}(\underline{w}_{\epsilon}):=\frac{1}{2}\int_{B_{\epsilon^{-1}r_{1}}}[|\nabla
\underline{w}_{\epsilon}|^{2}+\underline{w}_{\epsilon}^{2}]dx
-\frac{1}{p+1}\int_{B_{\epsilon^{-1}r_{1}}}[\min_{x\in
B_{r_{1}}}K(x)]\underline{w}_{\epsilon}^{p+1}dx \]
is non-increasing in $\epsilon$ as $\epsilon$ tends to 0. Since
$\underline{w}_{\epsilon}$ is a solution of (3.2), we have, by
multiplying (3.2) by $\underline{w}_{\epsilon}$ and integrating by parts,
\[ \int_{B_{\epsilon^{-1}r_{1}}\
}[|\nabla
\underline{w}_{\epsilon}|^{2}+\underline{w}_{\epsilon}^{2}]dx
=\int_{B_{\epsilon^{-1}r_{1}}}[\min_{x\in
B_{r_{1}}}K(x)]\underline{w}_{\epsilon}^{p+1}dx. \]
Hence
\begin{equation}
I_{\epsilon}(\underline{w}_{\epsilon})
=(\frac{1}{2}-\frac{1}{p+1})\int_{B_{\epsilon^{-1}r_{1}}} [|\nabla
\underline{w}_{\epsilon}|^{2}+\underline{w}_{\epsilon}^{2}]dx.
\end{equation}
>From the fact that $I_{\epsilon}(\underline{w}_{\epsilon})$ is
decreasing in $\epsilon$
and (3.4) we conclude that
$\{\underline{w}_{\epsilon} \}$ is bounded in
$W^{1,2}(R^{N})$. A standard boot-strapping argument shows that
$\{\underline{w}_{\epsilon} \}$ is bounded in
$C^{2,\alpha}_{loc}(R^{N})$. For the sake of completeness we
include the boot-strapping argument here.
Since $\{\underline{w}_{\epsilon} \}$ is bounded in
$W^{1,2}(R^{N})$, also in $L^{2^{*}}(B_{r})$ for all $r>0$ where
$2^{*}$ denotes the Sobolev conjugate of $2$, therefore
$\{\underline{w}_{\epsilon} \}$ is bounded in $W^{2,p_{1}}(B_{r-1})$
with $p_{1}=2^{*}/p>1$ by the interior elliptic $L^{p}$ regularity
theory (see, for example, \cite{gt}). Sobolev embedding
theorem implies that $\{\underline{w}_{\epsilon} \}$ is bounded in
$L^{p_{1}^{*}}(B_{r-1})$. If we use the above argument successively,
we can show that after $k=k(N,p)$ times,
$\{\underline{w}_{\epsilon} \}$ is bounded in $W^{2,p_{k}}(B_{r-k})$
with $p_{k}>N/2$. By the Sobolev embedding theorem
$\{\underline{w}_{\epsilon} \}$ is bounded in $C^{\alpha}(B_{r-k})$
for some $\alpha >0$. Now applying the interior Schauder estimate, we
conclude that $\{\underline{w}_{\epsilon} \}$ is bounded in
$C^{2,\alpha}(B_{r-k})$, hence bounded in $C^{2,\alpha}_{loc}(R^{N})$.
Passing to a subsequence if necessary, we have
\[ \underline{w}_{\epsilon} \rightarrow \underline{w}' \]
in $C^{2,\alpha}_{loc}(R^{N})$ where $\underline{w}'$ solves
\[ \left \{ \begin{array}{l}
\Delta u-u+[\min_{x\in B_{r_{1}}}K(x)]u^{p}=0 \\
u>0, \ \lim_{|x|\rightarrow \infty}u(x)=0, \ \nabla u(0)=0 .
\end{array} \right. \]
Notice that here we say $\underline{w}'>0$ because
\[ \underline{w}_{\epsilon}(0) \geq (\frac{1}{[\min_{x\in
B_{r_{1}}}K(x)]})^{\frac{1}{p-1}} \]
by the maximum principle. From the uniqueness result in \cite{k}, we
have
\[ \underline{w}'=[\min_{x\in B_{r_{1}}}K(x)]^{-\frac{1}{p-1}}w; \]
hence
\[ \underline{w}_{\epsilon} \rightarrow [\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{1}{p-1}}w \ \mbox{in} \
C^{2,\alpha}_{loc}(R^{N}). \]
To show
\[ \underline{w}_{\epsilon} \rightarrow [\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{1}{p-1}}w \ \mbox{in} \
W^{1,2}(R^{N}), \] we need only to show
\[ \overline{\lim_{\epsilon \rightarrow 0}} \|\underline{w}_{\epsilon}
\|_{W^{1,2}(R^{N})} \leq
\| [\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{1}{p-1}}w \|_{W^{1,2}(R^{N})}. \]
Using Lemma 2.1 again with the test function
\[ [\min_{x\in
B_{1}}K(x)]^{-\frac{1}{p-1}}w-[\min_{x\in B_{1}}K(x)]^{-\frac{\
1}{p-1}}w(\epsilon^{-1}r_{1}), \] we have
\[ I_{\epsilon}(w_{\epsilon}) \leq M[[\min_{x\in
B_{1}}K(x)]^{-\frac{1}{p-1}}w-[\min_{x\in
B_{1}}K(x)]^{-\frac{1}{p-1}}w(\epsilon^{-1}r_{1})]= \]
\[ \max_{t>0} \{ \frac{t^{2}}{2}\int_{B_{\epsilon^{-1}r_{1}}} [|\nabla
[\min_{x\in B_{1}}K(x)]^{-\frac{1}{p-1}}w|^{2}+|[\min_{x\in
B_{1}}K(x)]^{-\frac{1}{p-1}}w \]
\[ -[\min_{x\in B_{1}}K(x)]^{-\frac{1}{p-1}}w(\epsilon^{-1}r_{1})|^{2}]dx \]
\[ -\frac{t^{p+1}}{p+1}
\int_{B_{\epsilon^{-1}r_{1}}}[\min_{x\in
B_{r_{1}}}K(x)]|[\min_{x\in B_{1}}K(x)]^{-\frac{1}{p-1}}w \]
\[ -[\min_{x\in
B_{1}}K(x)]^{-\frac{1}{p-1}}w(\epsilon^{-1}r_{1})|^{p+1}dx \}. \]
It is easy to see that the maximum is obtained at
\[ t=t_{\epsilon}=
[\frac{\int_{B_{\epsilon^{-1}r_{1}}}[|\nabla
w|^{2}+|w-w(\epsilon^{-1}r_{1})|^{2}]}{\int_{B_{\epsilon^{-1}r_{1}}}
|w-w(\epsilon^{-1}r_{1})|^{p+1}}]^{1/(p-1)}. \]
By Proposition 2.2 (3), we see
\[ \lim_{\epsilon \rightarrow 0}t_{\epsilon}=[\frac{\int_{R^{N}}[|\nabla
w|^{2}+w^{2}]}{\int_{R^{N}}w^{p+1}}]^{1/(p-1)} = 1. \]
Using Proposition 2.2 (3) again, we have
\[ \lim_{\epsilon \rightarrow
0}M[[\min_{x\in B_{1}}K(x)]^{-\frac{1}{p-1}}w-[\min_{x\in B_{1}}K(x)]^{-\frac{1}{p-1}}w(\epsilon^{-1}r_{1})]=[\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{2}{p-1}}I(w); \]
hence
\[ \overline{\lim_{\epsilon \rightarrow
0}}I_{\epsilon}(\underline{w}_{\epsilon})\leq [\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{2}{p-1}}I(w), \]
and by (3.4)
\[ \overline{\lim_{\epsilon \rightarrow 0}} \|\underline{w}_{\epsilon}
\|_{W^{1,2}(R^{N})} \leq
\| [\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{1}{p-1}}w \|_{W^{1,2}(R^{N})}. \]
So we have proved the claim.
It follows from the claim that
\begin{eqnarray}
\lim_{\epsilon \rightarrow
0}\epsilon^{-N}\underline{J}_{\epsilon}(\underline{u}_{\epsilon})=[\min_{x\in B_{r_{1}}}K(x)]^{-\frac{2}{p-1}}I(w); \\
\lim_{\epsilon \rightarrow
0}\epsilon^{-N}\overline{J}_{\epsilon}(\overline{u}_{\epsilon})=[\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}}I(w).
\end{eqnarray}
Therefore (3.1) implies
\[ [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}}I(w) \leq
\liminf_{\epsilon \rightarrow
0}\epsilon^{-N}J_{\epsilon}(u_{\epsilon}) \]
\begin{equation} \leq \limsup_{\epsilon \rightarrow
0}\epsilon^{-N}J_{\epsilon}(u_{\epsilon}) \leq [\min_{x\in
B_{r_{1}}}K(x)]^{-\frac{2}{p-1}}I(w). \end{equation}
Now if we choose $O=O_{n} \in \Omega$ such that \[ O_{n} \rightarrow P
\in \overline{\Omega} \] where $\max_{x\in
\overline{\Omega}}K(x)=K(P)$ and choose
$r_{1}=r_{1n}=\frac{1}{n}\mbox{dist}(O_{n}, \partial \Omega )
\rightarrow 0$, then as $n \rightarrow \infty$, (3.7) implies
\[ \lim_{\epsilon \rightarrow
0}\epsilon^{-N}J_{\epsilon}(u_{\epsilon})=[\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}}I(w). \]
\ep
Next we turn our attention to the Neumann problem. We prove
\begin{proposition}
Letting $u_{\epsilon}$ be a least-energy solution of (1.1) with the
Neumann boundary condition, then we have the following:
\begin{enumerate}
\item If
\[ \max_{x\in \overline{\Omega}}K(x) > 2^{\frac{p-1}{2}}\max_{x\in
\partial \Omega}K(x), \] then
\[ \overline{\lim_{\epsilon \rightarrow
0}}\epsilon^{-N}J_{\epsilon}(u_{\epsilon}) \leq [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}}I(w). \]
\item If
\[ \max_{x\in \overline{\Omega}}K(x) <
2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x),\] then
\[ \overline{\lim_{\epsilon \rightarrow
0}}\epsilon^{-N}J_{\epsilon}(u_{\epsilon}) \leq \frac{1}{2}
[\max_{x\in \partial \Omega}K(x)]^{-\frac{2}{p-1}}I(w). \]
\end{enumerate}
\end{proposition}
\noindent {\it Proof of Part 1.} Let $P\in \Omega$ such that
$K(P)=\max_{x\in \overline{\Omega}}K(X)$. Notice that in this case
maximum of $K$ must be achieved in $\Omega$. Define
\[ w_{\epsilon}=u_{\epsilon}(\epsilon x+P) \] for $x\in
\Omega_{\epsilon}=\{x \in R^{N}:\ \epsilon x+P\in \Omega\}.$ Notice
that $\Omega_{\epsilon}$ is expanding toward $R^{N}$ as $\epsilon
\rightarrow 0$.
Let $ w_{\ast}(x)=[K(P)]^{-\frac{1}{p-1}}w$. Define
\[ I_{\epsilon}(v)=\frac{1}{2}\int_{\Omega_{\epsilon}} [|\nabla
v|^{2}+v^{2}]-\frac{1}{p+1}\int_{\Omega_{\epsilon}}K(\epsilon
x+P)v^{p+1}. \] Hence
\[ M_{\epsilon}(w_{\ast}):=\max_{t>0} I_{\epsilon}(tw_{\ast}) \]
\[ =\max_{t>0}\{ \frac{t^{2}}{2}\int_{\Omega_{\epsilon}}[|\nabla
w_{\ast}|^{2}+
w_{\ast}^{2}]-\frac{t^{p+1}}{p+1}\int_{\Omega_{\epsilon}}K(\epsilon
x+P)w_{\ast}^{p+1} \}. \]
Observe that the maximum for $t$ is obtained at
\[ t_{\epsilon}=[\frac{\int_{\Omega_{\epsilon}}|\nabla
w_{\ast}|^{2}+w_{\ast}^{2}}{\int_{\Omega_{\epsilon}} K(\epsilon
x+P)w_{\ast}^{p+1}} ]^{-\frac{1}{p-1}} \rightarrow 1 \]
by Proposition 2.2 (3). Using Proposition 2.2 (3) again, we have, with
the aid of the Lebesgue domination convergence theorem,
\[ \lim_{\epsilon \rightarrow 0}M_{\epsilon}(
w_{\ast})=\frac{1}{2}\int_{R^{N}}[|\nabla
w_{\ast}|^{2}+w_{\ast}^{2}]-\frac{1}{p+1}\int_{R^{N}}K(P)w_{\ast}^{P+1}
\]
\[ =[K(P)]^{-\frac{2}{p-1}}I(w). \]
But by Lemma 2.1, we have
\[
\epsilon^{-N}J_{\epsilon}(u_{\epsilon})=I_{\epsilon}(u_{\epsilon}(\epsilon
x+P)) \]
\[ \leq M_{\epsilon}(w_{\ast}) \rightarrow [K(P)]^{-\frac{2}{p-1}}I(w), \]
so we have
\[ \limsup_{\epsilon \rightarrow
0}\epsilon^{-N}J_{\epsilon}(u_{\epsilon})\leq
[K(P)]^{-\frac{2}{p-1}}I(w)=[\max_{x \in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}}I(w). \]
\ %%%%%%%%
\noindent {\it Proof of Part 2. } In this case we shall construct a
test function on the boundary of $\Omega$. Let $P\in \partial \Omega$
such that $K(P)= \max_{x \in \partial \Omega }K(x)$. Without the loss of
generality, we can assume $P=0$. Then we introduce a map $\Phi$ from a
neighborhood of $P=0$ to a neighborhood of $0$
flattening the boundary of $\Omega$ around $0$. We may suppose
\[ \Phi : \ {\cal U} \rightarrow (R^{N})^{+} \]
where ${\cal U}$ is a neighborhood of 0 in $\Omega$ and
$(D\Phi)_{ij}(0)=\delta_{ij}$. Then we let $\eta$ be a smooth cut-off
function in $(R^{N})^{+}$ which is $0$ outside $\Phi({\cal U})$ and
$1$ in a neighborhood of the origin in $(R^{N})^{+}$. Let
\[ u_{\epsilon}^{\ast}(x)= \left \{ \begin{array}{l}
[\max_{x \in \partial \Omega
}K(x)]^{-\frac{1}{p-1}}w(\epsilon^{-1}\eta(\epsilon
\Phi(x))\Phi(x)), \ \mbox{if} \
x\in {\cal U} \\
0 \ \mbox{in} \ \Omega \backslash {\cal U}. \end{array} \right. \]
We now define the test function
\[ w_{\epsilon}^{\ast}(x)= u_{\epsilon}^{\ast}(\epsilon x) \]
in $\Omega_{\epsilon}$ where $\Omega_{\epsilon}=\{x \in R^{N}: \
\epsilon x \in \Omega \}$. Then as in the proof of part 1, define
\[ I_{\epsilon}(v)=\frac{1}{2}\int_{\Omega_{\epsilon}} [|\nabla
v|^{2}+v^{2}]-\frac{1}{p+1}\int_{\Omega_{\epsilon}}K(\epsilon
x+P)v^{p+1}. \] Hence
\[ M_{\epsilon}(w_{\epsilon}^{\ast}):=\max_{t>0}
I_{\epsilon}(tw_{\epsilon}^{\ast}) \]
\[ =\max_{t>0}\{ \frac{t^{2}}{2}\int_{\Omega_{\epsilon}}[|\nabla
w_{\epsilon}^{\ast}|^{2}+
(w_{\epsilon}^{\ast})^{2}]-\frac{t^{p+1}}{p+1}\int_{\Omega_{\epsilon}}K(\epsilon
x+P)(w_{\epsilon}^{\ast})^{p+1} \}. \]
A careful analysis, like the formula (3.10) in the proof of
Proposition 3.3 \cite{nt}, of $w_{\epsilon}^{\ast}$ shows that
\[ t_{\epsilon}=[\frac{\int_{\Omega_{\epsilon}}|\nabla
w_{\epsilon}^{\ast}|^{2}+(w_{\epsilon}^{\ast})^{2}}{\int_{\Omega_{\epsilon}} K(\epsilon
x+P)(w_{\epsilon}^{\ast})^{p+1}} ]^{-\frac{1}{p-1}} \]
\[ \rightarrow
[\frac{\int_{(R^{N})^{+}}|[K(P)]^{-\frac{1}{p-1}}\nabla
w|^{2}+([K(P)]^{-\frac{1}{p-1}}w)^{2}}{\int_{(R^{N})^{+}}K(P)([K(P)]^{-\frac{1}{p-1}}w)^{p+1}}]^{-\frac{1}{p-1}}=1 \]
as $\epsilon \rightarrow 0$ where $t_{\epsilon}$ assumes the maximum
of $I_{\epsilon}(tw_{\epsilon}^{\ast})$. Then we conclude
with the aid of the analysis of $w_{\epsilon}^{\ast}$ that
\[ \lim_{\epsilon \rightarrow 0}M_{\epsilon}(
w_{\epsilon}^{\ast}) \]
\[ =\frac{1}{2}\int_{(R^{N})^{+}}[|
[K(P)]^{-\frac{1}{p-1}}\nabla
w|^{2}+([K(P)]^{-\frac{1}{p-1}}w)^{2}] \]
\[ -\frac{1}{p+1}\int_{(R^{N})^{+}}([K(P)]^{-\frac{2}{p-1}}w^{p+1}\]
\[ =\frac{1}{2}[K(P)]^{-\frac{2}{p-1}}I(w). \] But by Lemma 2.1, we
have
\[
\epsilon^{-N}J_{\epsilon}(u_{\epsilon})=I_{\epsilon}(u_{\epsilon}(\epsilon
x+P)) \]
\[ \leq M_{\epsilon}(w_{\epsilon}^{\ast}) \rightarrow \frac{1}{2}[K(P)]^{-\frac{2}{p-1}}I(w), \]
so we have
\[ \limsup_{\epsilon \rightarrow
0}\epsilon^{-N}J_{\epsilon}(u_{\epsilon})\leq\frac{1}{2}
[K(P)]^{-\frac{2}{p-1}}I(w)=\frac{1}{2}[\max_{x \in
\partial \Omega}K(x)]^{-\frac{2}{p-1}}I(w). \]
\ep
\begin{remark}
The inequalities in the conclusions of Proposition 3.2 are indeed
equalities. See Remark 5.1 after the proof of Theorem 1.3.
\end{remark}
\section{Proof of Theorem 1.1}
\setcounter{equation}{0}
\noindent {\it Proof of Part 1.}
Let $P_{\epsilon}$ be a local maximum of $u_{\epsilon}$ Then the maximum
principle implies
\[ u_{\epsilon}(P_{\epsilon}) \geq [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{1}{p-1}}. \]
>From Proposition 3.1 we know $w_{\epsilon}(x)=u_{\epsilon}(\epsilon
x+P_{\epsilon})$ is bounded in $W^{1,2}(R^{N})$. Then the standard
boot-strapping argument plus the elliptic regularity implies that
$w_{\epsilon}$ is uniformly bounded in $R^{N}$; hence $u_{\epsilon}$
is uniformly bounded in $\Omega$. This proves part 1.
\ep
\ %%%%%%%%%%%%%%%%
\noindent {\it Proof of Part 2.}
Suppose $\epsilon_{n}^{-1}\mbox{dist} (P_{\epsilon_{n}}, \partial \Omega )< R$
for a sequence $\epsilon_{n}$ with $\epsilon_{n} \rightarrow 0$ and
$R>0$ independent of $\epsilon_{n}$. We denote $P_{\epsilon_{n}}$ by
$P_{n}$ for simplicity. Let $Q_{n}\in \partial \Omega$ such that
$\mbox{dist}(P_{n},Q_{n})=\mbox{dist}(P_{n}, \partial \Omega )$. Take
$\Phi_{n}$ to be a map flattening $\partial \Omega$ around $Q_{n}$. So
$\Phi_{n}$ maps a neighborhood of $Q_{n}$ to a neighborhood of $0$ in
$(R^{N})^{+}:=\{x: \ x_{N}>0 \}$. We may further assume
$Q_{n}\rightarrow Q$. Then $u'_{n}(x)=u_{n}(\Phi_{n}^{-1}x)$ solves
\[ \frac{\epsilon^{2}}{2}
a^{(n)}_{ij}D^{ij}u+b^{(n)}_{i}D^{i}u-u+K(\Phi ^{-1}(x))u^{p}=0 \]
in a neighborhood of $0$ in $(R^{N})^{+}$
where $a^{(n)}_{ij}$ and $b^{(n)}_{i}$ depend on $n$ but their
$C^{\alpha}$ norm can
be bounded uniformly from both below and above and the ellipticity of
$a_{ij}$ are also bounded from below and above. Indeed we make
$\Phi_{n}$ appropriate so that $a^{(n)}_{ij}(0)=\delta_{ij}$ and
$b^{(n)}_{i}(0)=0$. See \cite{nt} section 4 for details.
Then $w_{n}(x)=u'_{n}(\epsilon x )$ solves
\[ a^{(n)}_{ij}(\epsilon x )D^{ij}u+b^{(n)}_{i}(\epsilon x
)D^{i}u-u+K(\Phi^{-1} (x))u^{p}=0
\]
in an expanding domain of $(R^{N})^{+}$. Letting $n \rightarrow
\infty$, we see $w_{n}\rightarrow w'$ in $C^{2,\alpha}_{loc}(R^{N})^{+})$
where $w'$ solves
\[ \left \{ \begin{array}{l} \Delta u -u+K(Q)u^{p}=0 \ \mbox{in} \
(R^{N})^{+} \\
u=0 \ \mbox{on} \ \{x=(x_{1},...,x_{N})\in R^{N}:\ x_{N}=0 \}.
\end{array} \right. \]
We observe that since
$\epsilon_{n}^{-1}\mbox{dist} (P_{\epsilon_{n}}, \partial \Omega )< R$,
$\epsilon_{n}^{-1}\Phi(P_{n})$ stays in $B_{R}(0)$. But $w_{n}$
gets a local maximum at $\epsilon_{n}^{-1}\Phi(P_{n})$, so
\[ w_{n}(\epsilon_{n}^{-1}\Phi(P_{n})) \geq [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{1}{p-1}}; \]
hence $w' \not \equiv 0$.
However since $w'$ has finite $W^{1,2}((R^{N})^{+})$ norm, $w'\equiv
0$ by Proposition 2.3. We reached a contradiction, so we have proved
\begin{equation} \lim_{\epsilon \rightarrow 0}\epsilon^{-1}\mbox{dist} (P_{\epsilon},
\partial \Omega )= \infty. \end{equation}
Assume $u_{\epsilon_{n}}$ has two local maximum points $P_{n}$ and
$P'_{n}$ for a sequence $\epsilon_{n} \rightarrow 0$. We first claim
\begin{equation} \lim_{n \rightarrow \infty}\epsilon^{-1}_{n}\mbox{dist}(P_{n},
P'_{n})=\infty.
\end{equation}
Again passing to a subsequence if necessary we suppose that there exist
$\{P_{n}\}$, $\{P_{n}'\}$ and $R$ independent of $n$ such that
\begin{equation} \epsilon^{-1}_{n}\mbox{dist}(P_{n}, P'_{n}) \leq R.
\end{equation}
As before we set $w_{n}(x)=u_{\epsilon_{n}}(\epsilon x+P_{n})$.
Because $w_{n}(x)$ solves
\[ \left \{ \begin{array}{l}
\Delta u-u+K(\epsilon_{n} x+P_{n})u^{p}=0 \ \mbox{in} \ \Omega_{n} \\
u|_{\Omega_{n}}=0
\end{array} \right. \]
where
\[ \Omega_{n} \rightarrow R^{N} \]
in the obvious sense, we conclude with the aid of part 1 and Proposition
3.1 that
\begin{equation} w_{n} \rightarrow [K(P)]^{-\frac{1}{p-1}}w
\end{equation} in $C^{2,\alpha}_{loc}(R^{N}) $
where $P$ is a limit point of $\{ P_{n} \}$. Since
$[K(P)]^{-\frac{1}{p-1}}w$ has only one critical point at $0$ which is
non-degenerate, $w_{n}$ can not have any other critical point except $0$ in
$B_{R}$. This contradicts (4.3), so we have proved (4.2).
Now consider the energy of $w_{n}$. Defining energy $I_{n}$ for $w_{n}$
by
\[ I_{n}(v)=\frac{1}{2}\int_{\Omega_{n}}[|\nabla v|^{2}+v^{2}]dx
-\frac{1}{p+1}\int_{\Omega_{n}}v^{P+1}_{+} dx, \] we have
\[
I_{n}(w_{n})=(\frac{1}{2}-\frac{1}{p+1})\|w_{n}\|_{W^{1,2}(\Omega_{n})}^{2}
\]
\[ \geq
(\frac{1}{2}-\frac{1}{p+1})(\|w_{n}\|_{W^{1,2}(B_{1})}^{2}+
\|w_{n}\|_{W^{1,2}(B_{2})}^{2})\]
where $B_{1}=B_{r_{n}}((P_{n})$,
$ B_{2}=B_{r_{n}}(P_{n}')$ and
$r_{n}=\frac{\mbox{dist}(P_{n},P'_{n})}{2\epsilon_{n}}$. Since $r_{n}
\rightarrow \infty$, repeating the construction leading to (4.4) for
both $P_{n}$ and $P_{n}'$, we have
\[ w_{n}(\epsilon_{n}^{-1}x+P_{n})|_{B_{1}} \rightarrow
[K(P)]^{-\frac{1}{p-1}}
w; \]
\[ w_{n}(\epsilon_{n}^{-1}x+P_{n}')|_{B_{2}} \rightarrow
[K(P')]^{-\frac{1}{p-1}}w \]
in $C_{loc}^{2,\alpha}(R^{N})$ where $P'$ is a limit point of $P_{n}'$. Hence
\[ \liminf_{n \rightarrow \infty}\|w_{n}\|_{W^{1,2}}^{2} \geq
2\|[K(P)]^{-\frac{1}{p-1}}w\|_{W^{1,2}}^{2}. \]
Therefore
\[ \liminf_{n \rightarrow
\infty}\epsilon_{n}^{-N}J_{\epsilon_{n}}(u_{\epsilon_{n}})
=\liminf_{n \rightarrow \infty}I_{n}(w_{n}) \]
\[ \geq 2(\frac{1}{2}-\frac{1}{p}) \|w\|_{W^{1,2}}^{2}=2 [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-2}}I(w) \]
which contradicts Proposition 3.1. The proof of part 2 is now
complete. \ep
\ %%%%%%%%
\noindent {\it Proof of Part 3. } Suppose that there is a sequence
$\{ P_{\epsilon_{n}} \}$ of $\{ P_{\epsilon} \}$ with
\[ P_{n} :=P_{\epsilon_{n}} \rightarrow P \]
where $K(P) < \max_{x \in \overline{\Omega}}K(x) $. Consider
\[ w_{n}(x):=u_{\epsilon_{n}}(\epsilon_{n}x+P_{n}) \]
in $\Omega_{n}=\{x \in R^{N}: \ \epsilon_{n}x+P_{n}\in \Omega \}$.
>From part (2) of this theorem, we know
\[ \Omega_{n} \rightarrow R^{N}. \]
Then a boot-strapping argument again shows that
\[ w_{n} \rightarrow w' \]
in $C^{2,\alpha}_{loc}(R^{N})$ and $w'$ is a positive solution of
\[ \Delta u -u + K(P)u^{p}=0 \]
which decays at infinity. Therefore the uniqueness result in \cite{k}
implies that $w'=[K(P)]^{-\frac{1}{p-1}}w$. Hence
\[ w_{n} \rightarrow [K(P)]^{-\frac{1}{p-1}}w \]
weakly in $W^{1,2}(R^{N})$ by the Fatou's lemma, so
\[ \liminf_{n \rightarrow \infty}\epsilon^{-N}
J_{\epsilon_{n}}(u_{\epsilon_{n}})=\liminf_{n \rightarrow
\infty}(\frac{1}{2}-\frac{1}{p+1})\|w_{n}\|^{2}_{W^{1,2}(R^{N})} \]
\[ \geq (\frac{1}{2}-\frac{1}{p+1})
\|[K(P)]^{-\frac{1}{p-1}}w\|^{2}_{W^{1,2}(R^{N})}
=[K(P)]^{-\frac{2}{p-1}} I(w) \]
\[ > [\max_{x \in \overline{\Omega}}K(x)]^{-\frac{2}{p-1}} I(w). \]
But this is inconsistent with Proposition 3.1, so we have proved part
3. \ep
\begin{remark} If function $K(x)$ has only one local maximum point $P$
on the boundary of $\Omega$, then the peaks of least-energy solutions
have to converge to $P$ by Theorem 1.1 (3). But from Theorem 1.1 (2),
we also know that those peaks converge to $P$ {\em slowly}.
\end{remark}
\section{Proof of Theorem 1.2 and 1.3}
\setcounter{equation}{0}
\noindent {\it Proof of Theorem 1.2.}
The proof of Part 1 is identical with the proof of part 1 of Theorem
1.1. One applies the maximum principle to get a lower bound for
$u_{\epsilon}$ and the boot-strapping argument to get an upper bound
for $u_{\epsilon}$. We leave the details to reader.
To prove part 2, we first prove that $P_{\epsilon}$ stays away from
the boundary of $\Omega$. Suppose $P_{\epsilon_{n}} \rightarrow P\in
\partial \Omega$ for a sequence $\epsilon_{n}$. We consider two cases. First
assume
\[ \lim_{\epsilon_{n} \rightarrow
0}\epsilon_{n}^{-1}\mbox{dist}(P_{\epsilon_{n}}, P)= \infty. \]
Then with the aid of a boot-strapping argument as in the proof of
Proposition 3.1,
we know
\[ u_{\epsilon_{n}}(\epsilon_{n}x+P_{\epsilon_{n}}) \rightarrow
[K(P)]^{-\frac{1}{p-1}}w \]
in $C_{loc}^{2,\alpha}(R^{N})$. Hence
\[ \liminf_{\epsilon_{n} \rightarrow
0}\epsilon_{n}^{-N}J_{\epsilon_{n}}( u_{\epsilon_{n}}) \geq
[K(P)]^{-\frac{2}{p-1}} I(w) > [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}} I(w) \]
which contradicts Proposition 3.2. Here the last strict inequality
follows from the assumption
\[ \max_{x\in \overline{\Omega}
}K(x) > 2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x). \]
Next we assume
\[ \epsilon_{n}^{-1}\mbox{dist}(P_{\epsilon_{n}}, P) \leq R \]
for some $R>0$ independent of $\epsilon_{n}$. Again by the
boot-strapping technique, we know
\[ u_{\epsilon_{n}}(\epsilon_{n}x+P_{\epsilon_{n}}) \rightarrow
[K(P)]^{-\frac{1}{p-1}}w \]
in $C_{loc}^{2,\alpha}((R^{N})^{+})$. Hence
\[ \liminf_{\epsilon_{n} \rightarrow
0}\epsilon_{n}^{-N}J_{\epsilon_{n}}( u_{\epsilon_{n}}) \geq
\frac{1}{2}[K(P)]^{-\frac{2}{p-1}} I(w) > [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}} I(w) \]
which again contradicts Proposition 3.2. The last strict inequality is
exactly the assumption
\[ \max_{x\in \overline{\Omega}
}K(x) > 2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x). \]
Therefore all local maximum
of $u_{\epsilon}$ stays away from the boundary of $\Omega$.
Now we show that $u_{\epsilon}$ has only one local maximum for
$\epsilon$ small enough. Suppose $P_{\epsilon_{n}}$ and
$P_{\epsilon_{n}}'$ are two local maximum for a subsequence
$\{\epsilon_{n}\}$. As in (4.2), we know
\[ \lim_{\epsilon_{n} \rightarrow
0}\epsilon_{n}^{-1}\mbox{dist}(P_{\epsilon_{n}}, P_{\epsilon_{n}}')
=\infty . \]
If
\[ \lim_{\epsilon_{n} \rightarrow
0}\epsilon_{n}^{-1}\mbox{dist}(P_{\epsilon_{n}}, P_{\epsilon_{n}}')
=\infty , \]
we can proceed as in the proof of Theorem 1.1 part 2.
Consider the energy of $u_{\epsilon_{n}}$ and conclude
\[ \liminf_{\epsilon_{n} \rightarrow
0}J_{\epsilon_{n}}(u_{\epsilon_{n}}) \geq 2[\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}} I(w) \]
which contradicts Proposition 3.2. So we proved part 2.
Finally we let $P\in \Omega$ be a limit point of
$\{P_{\epsilon_{n}}\}$. Then since
\[ u_{\epsilon_{n}}(\epsilon_{n}x+P_{\epsilon_{n}}) \rightarrow
[K(P)]^{-\frac{1}{p-1}}w \]
in $C_{loc}^{2,\alpha}((R^{N})^{+})$, we have by the Fatou lemma
\begin{equation} \liminf_{\epsilon_{n}\rightarrow
0}J_{\epsilon_{n}}(u_{\epsilon_{n}}) \geq [K(P)]^{-\frac{2}{p-1}}
I(w) .\end{equation}
It follows from Proposition 3.2 that
\[ K(P)=\max_{x\in \overline{\Omega}}K(x) .\]
\ep
\ %%%%%%%%%%%%%%%
\noindent {\it Proof of Theorem 1.3.}
The proof of this theorem is similar to the proof of Theorem 2.1
\cite{nt}, so we shall be very sketchy. The proof of part 1 is again
identical with the proof of part 1 of Theorem 1.1. We omit it.
To prove part 2, we first show that for a local maximum $P_{\epsilon}$ of $u_{\epsilon}$
\[ \epsilon^{-1}\mbox{dist}(P_{\epsilon}, \partial \Omega)
2^{\frac{p-1}{2}}\max_{x\in
\partial \Omega}K(x), \]
\[ \liminf_{\epsilon \rightarrow
0}\epsilon^{-N}J_{\epsilon}(u_{\epsilon}) \geq [\max_{x\in
\overline{\Omega}}K(x)]^{-\frac{2}{p-1}}I(w), \]
and when
\[ \max_{x\in \overline{\Omega}}K(x) <
2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x), \]
\[ \liminf_{\epsilon \rightarrow
0}\epsilon^{-N}J_{\epsilon}(u_{\epsilon}) \geq \frac{1}{2}
[\max_{x\in \partial \Omega}K(x)]^{-\frac{2}{p-1}}I(w). \]
So the inequalities in Proposition 3.2 are actually equalities with
$\overline{\lim}$ replaced by $\lim$.
\end{remark}
\begin{remark}
The border line case $\max_{x\in \overline{\Omega}}K(x) =
2^{\frac{p-1}{2}}\max_{x\in \partial \Omega}K(x)$ for the Neumann
problem seems quite interesting to me. I don't know if the geometry of
$\Omega$ will come out and affect the location of the peaks in this
case.
\end{remark}
\ %%%%%%%%
\noindent {\bf Acknowledgement} This work is part of the author's Ph.D
thesis at University of Minnesota. He gratefully acknowledges the
encourage received from his adviser, Professor Wei-Ming Ni, in the
course of this work.
\begin{thebibliography}{99}
\bibitem{ar} A. Ambrosetti and P. H. Rabinowitz, Dual variational
methods in critical point theory and applications. J. Funct. Anal. 14,
1973, pp. 349-381.
\bibitem{el} M. J. Esteban and P. J. Lions, Existence and non-existence
results for semilinear elliptic problems in unbounded domains,
Proceedings of the Royal Society of Edinburgh, 93, 1982.
\bibitem{gnn} B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry and
related Properties via the maximum principle, Comm. Math. Phys.(68)
no.3, 1979, 209-243.
\bibitem{gm} A. Gierer and H. Meinhardt, A theory of biological
pattern formation, Kybernetik (Berlin) 12. 1972, pp. 30-39.
\bibitem{gt} D. Gilbarg and S. N. Trudinger, Elliptic Partial
Differential Equations of Second Order, Second Edition,
Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983.
\bibitem{ks} E. Keller and L. Segel, Initiation of slime mold
aggregation viewed as an instability, J. Theo. Biol. 26, 1970, pp. 399-415.
\bibitem{k} M. K. Kwong, Uniqueness of positive solutions of $\Delta
u-u+u^{p}=0$ in $R^{n}$, Arch. Rational Mech. Anal. 105, 1989.
\bibitem{lnt} C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude
stationary solutions to a chemotaxis system, J. Diff. Equa., 72, 1988,
pp. 1-27.
\bibitem{npt} W.-M. Ni, X. Pan, and I. Takagi, Singular behavior of
least-energy solutions of a semilinear Neumann problem involving
critical Sobolev exponents, Duke Math. J. Vol 67, No. 1, 1992.
\bibitem{nt} W.-M. Ni and I. Takagi, On the shape of least-energy
solutions to a semilinear Neumann Problem, Comm. Pure. Appl. Math.,
Vol. XLIV, no. 7, 1991.
\bibitem{nt2} W.-M. Ni and I. Takagi, Locating the peaks of
least-energy solutions to a semilinear
Neumann problem, Duke Math. J. to appear.
\bibitem{p}X. Pan, Condensation of least-energy solutions of a
semilinear Neumann problem, J. P.D.E. to appear.
\bibitem{rw} X. Ren and J. Wei, On a two dimensional elliptic problem
with large exponent in nonlinearity, Tran. A.M.S., to appear.
\bibitem{rw2} X. Ren and J. Wei, Counting peaks of solutions to some
quasilinear elliptic equations with large exponent, J. Diff. Equa., to
appear.
\end{thebibliography}
{\sc School of Mathematics, University of Minnesota,
127 Vincent Hall, 206 Church Street S.E.
Minneapolis, MN 55455}\par
E-mail address: ren\@@s5.math.umn.edu
\end{document}